Dyadic Division Towers of the Angle Difference in the Pythagorean Angle Lattice: Cyclotomic Descent, Exact Decoding, Power Resonance and Arithmetic Moduli
Year: 2026
Type: preprint
Abstract: Dyadic division towers of the angle difference in the Pythagorean Angle Lattice are completely classified by cyclotomic descent, exact address decoding, power resonance classes, and arithmetic moduli of Gaussian exponent vectors. Let θ be the smaller acute angle of a primitive Pythagorean triangle T = (a, b, c) and let Δ = π/2 − 2θ be the difference of its two acute angles. Repeated halving of Δ produces the real fields F_r = ℚ(cot(Δ/2^r)) and the complex fields E_n = ℚ(i, e^{iΔ/2^n}) = F_{n+1}(i). We give an exact arithmetic classification of this dyadic division tower. The fields F_r have degree 2^{r−1}, are totally ramified at 2, and F_3 = ℚ(√2, √(c(2 + √2))) depends only on the hypotenuse. Writing Δ = 4 ∑_p e_p φ_p + (π/2) L, where e_p are the oriented Gaussian prime exponents of the source of T, φ_p are fixed Gaussian prime arguments and L is an odd integer (the sector coordinate, L ≡ −1 (mod 4) exactly when the smaller leg is odd), we prove that E_n ∩ ℚ(μ_{2^∞}) = ℚ(ζ_{2^{min(n+2,ν+3)}}), where 2^ν is the dyadic content of the exponent vector; E_n is Galois over ℚ(i) if and only if n ≤ ν + 3; two towers have K-conjugate n-th fields exactly when their exponent vectors agree modulo 2^{n−1} up to an odd scalar whose class modulo 4 is the product of their sector characters; and the actual embedded fields coincide exactly when the normalised addresses y = −2^{−ν} e / L agree modulo 2^{n−ν−1}. Consequently two complete towers coincide if and only if Δ′/Δ ∈ ℚ, and every such ratio has odd numerator and denominator. The difference angles of a resonance class are precisely the odd multiples of a fundamental angle below π/2; for instance (3, 4, 5), (44, 117, 125) and (237, 3116, 3125) have Δ′ = 3Δ and 5Δ and identical towers. We characterise exactly which addresses occur, count the triples above each address, describe the dyadic completion of the address space, and interpret the results for exact dyadic roots of rational phase gates. All field equalities refer to subfields of a fixed algebraic closure containing a fixed copy of ℚ(i); classification up to conjugacy is treated separately.
Author: Rajput, Chetansing
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221459797
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